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Assessing the effects of stochastic perception error
under travel time variability
Xiangdong Xu •Anthony Chen •Lin Cheng
Published online: 13 July 2012
Springer Science+Business Media, LLC. 2012
Abstract Perceived mean-excess travel time is a new risk-averse route choice criterion
recently proposed to simultaneously consider both stochastic perception error and travel
time variability when making route choice decisions under uncertainty. The stochastic
perception error is conditionally dependent on the actual travel time distribution, which is
different from the deterministic perception error used in the traditional logit model. In this
paper, we investigate the effects of stochastic perception error at three levels: (1) individual
perceived travel time distribution and its connection to the classification by types of
travelers and trip purposes, (2) route choice decisions (in terms of equilibrium flows and
perceived mean-excess travel times), and (3) network performance measure (in terms of
the total travel time distribution and its statistics). In all three levels, a curve fitting method
is adopted to estimate the whole distribution of interest. Numerical examples are also
provided to illustrate and visualize the above analyses. The graphical illustrations allow for
intuitive interpretation of the effects of stochastic perception error at different levels. The
analysis results could enhance the understanding of route choice behaviors under both
(subjective) stochastic perception error and (objective) travel time uncertainty. Some
suggestions are also provided for behavior data collection and behavioral modeling.
Keywords Mean-excess travel time Stochastic perception error
Travel time variability Moment analysis
X. Xu L. Cheng
School of Transportation, Southeast University, Nanjing 210096, People’s Republic of China
X. Xu A. Chen (&)
Department of Civil and Environmental Engineering, Utah State University, Logan,
UT 84322-4110, USA
e-mail: anthony.chen@usu.edu
A. Chen
Key Laboratory of Road and Traffic Engineering, Tongji University, Shanghai 201804,
People’s Republic of China
123
Transportation (2013) 40:525–548
DOI 10.1007/s11116-012-9433-6
Abbreviations
CL Confidence level
COV Coefficient of variation
METE Mean-excess traffic equilibrium
METT Mean-excess travel time
PBT Perceived buffer time
PEED Perceived expected excess delay
PMETT Perceived mean-excess travel time
PMT Perceived mean time
PTT Perceived travel time
PTTB Perceived travel time budget
SMETE Stochastic mean-excess traffic equilibrium
SPE Stochastic perception error
TTB Travel time budget
TTD Travel time distribution
TTT Total travel time
TTTB Total travel time budget
VMR Variance-mean ratio
Introduction
Recent empirical studies have revealed that travel time variability plays an important role
in travelers’ route choice decisions (Abdel-Aty et al. 1995; Brownstone et al. 2003;de
Palma and Picard 2005; Lam and Small 2001; Liu et al. 2004; Senna 1994; van Lint et al.
2008). Travelers treat travel time variability as a risk in their route choices, because it
introduces uncertainty for an on-time arrival at the destination. Due to its importance,
modeling route choice decisions under uncertainty is receiving great attention. Some of the
recent principal advances on this topic are classified in Table 1, which highlights the
aspects of travel time variability in consideration (e.g., reliability, unreliability, or both),
and the type of perception error (e.g., no perception error, deterministic perception error, or
stochastic perception error (SPE)). For other traffic equilibrium models under uncertainty,
interested readers may refer to the disutility/utility-based model (e.g., Chen et al. 2002;Di
et al. 2008; Mirchandani and Soroush 1987; Yin and Ieda 2001), game theory-based model
(e.g., Bell 2000; Bell and Cassir 2002; Szeto et al. 2006), and prospect theory-based model
(e.g., Connors and Sumalee 2009; Xu et al. 2011a,b).
Table 1 Classification of traffic equilibrium models under uncertainty (Chen and Zhou 2009; Chen et al.
2011)
Perception error
No Yes
Deterministic Stochastic
Travel time variability Travel time reliability TTB STTB –
Travel time unreliability LAPUE – –
Travel time reliability & unreliability METE – SMETE
526 Transportation (2013) 40:525–548
123
On the travel time variability aspect under consideration:
(1) TTB-type models use travel time budget (TTB) as a risk-averse route choice criterion.
TTB is defined as a travel time reliability chance constraint such that the probability of
completing a trip within the threshold is not less than a user-specified confidence level
a(Lo et al. 2006; Shao et al. 2006a,b). It is composed of the mean travel time plus a
buffer time, similar to the concept of effective travel time used in Uchida and Iida
(1993) to ensure the travel time reliability requirement. However, it does not account
for the unreliability aspect of travel time variability when the travelers are late (i.e.,
encountering worse travel times beyond the TTB in the distribution tail of 1 -a).
(2) The late arrival penalized user equilibrium (LAPUE) model proposed by Watling
(2006) considers the unreliability aspect of travel time variability (via a schedule
delay term) to penalize the late arrival for a fixed departure time (or a fixed TTB).
(3) The mean-excess traffic equilibrium (METE) model uses the mean-excess travel time
(METT) proposed by Chen and Zhou (2010) to describe travelers’ route choice
decisions under uncertainty. It is considered as a more complete and accurate
measure, because it simultaneously considers both reliability (on-time arrival) and
unreliability (late arrival) aspects of travel time variability to address two
fundamental questions, i.e., ‘‘how long do I need to allow for this trip’’ a n d ‘‘ how
bad should I expect for the worse cases’’ .
On the type of perception error aspect under consideration:
(1) The reliability-based stochastic user equilibrium (RSUE) model proposed by Shao et al.
(2006b) and the stochastic travel time budget equilibrium (STTBE) model proposed by
Siu and Lo (2006) consider travelers’ perception error on network conditions in the TTB
modeling framework. Thus, we can classify them as a stochastic travel time budget
(STTB) model. This consideration avoids the unrealistic assumption in the deterministic
TTB model that all travelers can accurately perceive the TTBs of all routes available.
However, the perception error herein is independent of the actual travel time distribution
(i.e., simply adding a random error term such as the Gumbel distribution to the TTB to
obtain the logit-based TTB model). Thus, the STTB models also inherit the drawbacks
of the logit-type SUE model. To facilitate differentiation, the perception error herein is
also referred to as a ‘‘deterministic perception error’’. Mirchandani and Soroush (1987)
stated that the deterministic perception error may not well reflect travelers’ perception
on the travel time distribution (TTD). They suggested using a stochastic perception error
(SPE), which is conditional on the actual (or random) TTD. In that case, incorporating
the SPE into the actual TTD forms the perceived TTD. Consequently, travelers make
route choice decisions based on the perceived TTD rather than the actual TTD.
(2) The stochastic mean-excess traffic equilibrium (SMETE) model embeds the SPE
concept into the METE modeling framework (Chen and Zhou 2009; Chen et al. 2011).
In this new model, the perception error is dependent on the actual TTD, which requires
first deriving the perceived TTD as shown in Fig. 1. Then, the perceived mean-excess
travel time (PMETT) concept is used as a new risk-averse criterion to explicitly account
for both reliability and unreliability aspects of travel time variability as well as traveler’s
knowledge on network conditions (via the SPE) in making route choice decisions.
Recently, several empirical studies examined ways to collect travelers’ perception on
travel time variability and found that the perception of unreliability plays an important role in
making route choice decisions under uncertainty. Tseng et al. (2009) conducted a pilot study
on the perception of travel time unreliability using a face-to-face in-depth interview to assess
Transportation (2013) 40:525–548 527
123
how travelers perceive travel time unreliability. Zhu and Srinivasan (2010) used the 2001
national household travel survey data to examine travelers’ perception on both congestion
and unreliability. Peer et al. (2010) compared the actual travel time distribution experienced
by the travelers to the perceived travel time distribution stated by the travelers in a survey.
Our study is different from the above empirical studies, because we provide an equilibrium
model-driven analysis of travelers’ route choice behaviors under both (subjective) stochastic
perception error and (objective) travel time uncertainty (namely the stochastic mean-excess
traffic equilibrium model). In contrast, the existing relevant literature mainly, if not all of
them, adopted empirical data-based statistical analyses to examine the case-specific route
choice characteristics based on either observed or simulated data. Also note that the concept
of stochastic perception error (SPE) and the risk-averse route choice criterion of perceived
mean-excess travel time (PMETT) are new theoretical developments, which have not been
explored in depth. How the SPE influences the perceived travel time distributions and route
choice decisions has not been investigated. In addition, how to interpret the SPE in realistic
trip decisions and how the SPE impacts on network-wide performance are useful for trav-
elers’ trip planning and planners’ network performance assessment.
Hence, the purpose of this study is to investigate the effects of SPE under travel time
variability at three levels: (1) individual perceived travel time distribution as well as its
connection to the classification by types of travelers and trip purposes, (2) PMETT-based
route choice decisions (in terms of equilibrium traffic flows and PMETTs), and (3) net-
work performance measure (in terms of the total travel time distribution and its statistics).
In all three levels, a curve fitting method is adopted to estimate the whole distribution of
interest based on its first four moments. Note that the moment analysis approach proposed
by Chen and Zhou (2009) and Chen et al. (2011) only provides the moments (e.g., mean,
variance, skewness, and kurtosis) of the perceived travel time. The moment information is
used with two asymptotic expansions to estimate the PMETT as a risk-averse route choice
criterion in the SMETE model. In order to enhance the understanding of the effect of SPE
and also the visualization on perceived travel time distribution, we use the fitting distri-
bution method to fit the moments obtained from the moment analysis approach in this
paper. The fitting distribution method is the main tool for visualizing the effect of SPE.
Distribution curves and statistics (or moments) are used simultaneously to complement
each other. Distribution curves provide a more intuitive ‘picture’ of distribution, and
statistics provide a more quantitative characterization of distribution.
Travel Time
Probability Density
Perceived Travel Time Distribution
Actual Travel Time Distribution
Fig. 1 Illustration of actual and perceived travel time distributions
528 Transportation (2013) 40:525–548
123
The remainder of this paper is organized as follows. ‘‘Mathematical formulation’’ section
presents the definition of PMETT and the SMETE conditions. ‘‘Fitting distribution by
moments’’ section provides a numerical method that makes use of the first four moments to fit
the whole distribution. This fitting distribution method is then used to investigate the effects
of SPE under travel time variability at three levels in ‘‘Numerical examples’’ section.
‘‘ Discussions’’ section discusses the interpretation of the SPE distribution and some impli-
cations of this study. Finally, some concluding remarks and future research directions are
provided in ‘‘Conclusions and future research’’ section.
Mathematical formulation
In this section, we present the definitions of PMETT and the SMETE conditions. Consider
a strongly connected network G=[N,A], where Nand Adenote the sets of nodes and
links, respectively. Let Wdenote the set of origin–destination (O–D) pairs for which travel
demand q
w
is generated between O–D pair w, and let fw
pdenote the flow on route p2Pw,
where P
w
is the set of routes connecting O–D pair w2W.
Definition 1 (Chen and Zhou 2009; Chen et al. 2011): The Perceived Travel Time ~
Tw
pon
route pbetween O–D pair wis defined as the sum of the actual (or random) travel time Tw
p
and the stochastic perception error ew
pTw
p
, i.e.,
~
Tw
p¼Tw
pþew
pTw
p
;8p2Pw;w2W;ð1Þ
where ew
pTw
p
denotes the SPE conditional on the actual travel time.
Definition 2 (Chen and Zhou 2009; Chen et al. 2011): The Perceived Mean-Excess
Travel Time ~
gw
paðÞon route pbetween O–D pair wwith respect to a predefined confidence
level ais defined as the conditional expectation of the perceived route travel time ~
Tw
p
exceeding the corresponding perceived route travel time budget ~
nw
paðÞ, i.e.,
~
gw
paðÞ¼E~
Tw
p~
Tw
p~
nw
paðÞ
hi
;8p2Pw;w2W;ð2Þ
where E½is the expectation operator and ~
nw
paðÞis defined as:
~
nw
paðÞ¼min ~
nPr ~
Tw
p~
n
a
no
¼E~
Tw
p
hi
þ~
cw
paðÞ;8p2Pw;w2W;ð3Þ
where ~
cw
paðÞis a perceived buffer time (PBT) added to the perceived mean time (PMT)
E~
Tw
p
hi
to ensure the reliability requirement for on-time arrivals at a confidence level a.
Meanwhile, Eq. (2) can also be rewritten as:
~
gw
pðaÞ¼~
nw
pðaÞþE~
Tw
p~
nw
pðaÞ
~
Tw
p~
nw
pðaÞ
hi
¼E~
Tw
p
hi
|fflfflffl{zfflfflffl}
perceived
mean time
þ~
cw
pðaÞ
|fflffl{zfflffl}
perceived
buffer time
þE~
Tw
p~
nw
pðaÞ
~
Tw
p~
nw
pðaÞ
hi
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
perceived expected excess delay
;8p2Pw;w2W;ð4Þ
Transportation (2013) 40:525–548 529
123
where the sum of the first two terms (i.e., PTTB) and the third term (i.e., the perceived
expected excess delay (PEED)) represent the reliability and unreliability aspects of the
perceived travel time variability, respectively. The components of PMETT are illustrated
in Fig. 2. PMETT explicitly captures the left region with the a-percentile of perceived
travel time reliability requirement and the right region with the (1 -a) percentile of
unreliability in the perceived travel time distribution tail. In this sense, PMETT explicitly
considers both on-time arrival (via PTTB) and late arrival (via PEED) as well as travelers’
SPE on network conditions. Thus, PMETT can be regarded as a more complete measure to
describe travelers’ route choice decisions under network uncertainty and perception error.
Definition 3 (Chen and Zhou 2009; Chen et al. 2011): The Stochastic Mean-Excess
Traffic Equilibrium (SMETE)is a network state such that for each O–D pair, all used
routes have equal PMETT, which is not larger than that on any unused route connecting
this O–D pair.
The above descriptive definition can be mathematically represented as follows:
~
gw
pf
ðÞ ¼~
uw;if fw
p[0
~
uw;if fw
p¼0
(;8p2Pw;w2W;ð5Þ
where ~
uwis the minimum PMETT between O–D pair w, i.e., ~
uw¼min ~
gw
p;p2Pw
no
. The
final exhibition of the above equilibrium state is that no traveler can improve his/her PMETT
by unilaterally changing his/her route choice. For the equivalent variational inequality (VI)
formulation, please refer to Appendix 1. In addition, the actual and perceived travel time
distributions are explicitly derived from the travel demand and SPE distributions. This
approach is thus different from the empirical studies on the perception of travel time unre-
liability using stated preference surveys (e.g., Tseng et al. 2009). For a detailed derivation of
the probability distribution statistics for the case where the uncertainty is induced by the
lognormal distributed travel demand, interested readers may refer to Appendix 2.
0 5 10 15 20 25 30
0
0.05
0.1
Perceived Travel Time
Probability Density
0 5 10 15 20 25 30
0
0.5
1
Perceived Travel Time
Cumulative Probability
PMT : perceived mean time
PB T : perce ived buffer t ime
PTTB : percei ved travel tim e budget
PEED : perceived expected excess delay
PMETT : perc eived me an-exce ss travel ti me
PMT PBT
PMETT
PEED
PTTB
Fig. 2 Illustration of the components of PMETT
530 Transportation (2013) 40:525–548
123
Fitting distribution by moments
Note that the moment analysis approach proposed by Chen and Zhou (2009) and Chen et al.
(2011) only provides the moments (e.g., mean, variance, skewness, and kurtosis) of the per-
ceived travel time to estimate the PMETT for the SMETE model. In this section, we describe a
numerical method to estimate the whole perceived travel time distribution. In order to enhance
the understanding of the effect of SPE and also the visualization on perceived travel time
distribution, we use the fitting distribution method to fit the moments obtained from the
moment analysis approach. Distribution curves and statistics are used simultaneously to
complement each other. Distribution curves provide a more intuitive ‘picture’ of distribution,
and statistics provide a more quantitative characterization of distribution. The fitting distri-
bution method is the main tool for visualizing the effect of SPE. Hence, the analysis purposes
and analysis tool are both different from Chen and Zhou (2009) and Chen et al. (2011).
Hill et al. (1976) proposed a numerical method that makes use of the first four moments
of a random variable to match any distribution in the Johnson curve system. Clark and
Watling (2005) adopted the same method to estimate the full distribution of total travel time.
Empirical study revealed that TTD is typically asymmetric with a long and fat tail. In this
study, we use the lognormal distribution to estimate the perceived distribution of interest.
Recall that from the moment analysis of travel time variables, we have obtained the four
commonly used statistics, i.e., mean (E), variance (Var), skewness (S), and kurtosis (K).
Consider the following lognormal random variable X:
Z¼cþdln XnðÞ;X[n;ð6Þ
where Zis a standard normal variable, i.e., Z*N(0, 1), and c,n, and dare parameters.
Then, the probability density function (PDF) of Xis as follows:
pdf XðÞ¼ 1
ffiffiffiffiffiffi
2p
pexp cþdln XnðÞðÞ
2
2
"#
d
Xn;X[n:ð7Þ
The remaining work is to calculate the values of the three parameters. First, we solve the
following equation to obtain x
ðx1Þðxþ2Þ2¼S2:ð8Þ
The three parameters in Eq. (7) can then be calculated as follows:
d¼ðln xÞ0:5;ð9Þ
c¼0:5dln xx1ðÞ
Var
;ð10Þ
n¼Eexp 1=ð2dÞc
d
:ð11Þ
After calibrating the above parameters, the PDF in Eq. (7) is actually known. Thus, an
estimated distribution of the random variable Xcan be obtained.
Numerical examples
In this section, we demonstrate the effects of SPE at three levels: (1) individual perceived
travel time distribution (TTD), (2) route choice decisions (in terms of equilibrium route
Transportation (2013) 40:525–548 531
123
flows and route PMETTs), and (3) network performance measure (in terms of the total
travel time distribution and its statistics).
For demonstration purpose, a simple network with one O–D pair and three parallel
links/routes is used to conduct a set of numerical experiments. The standard BPR function
TaVa
ðÞ¼t0
a1þ0:15 Va=Ca
ðÞ
4
hi
is adopted. The free-flow travel times of the three links
are, respectively, 22, 24, and 17 min, and their capacities are, respectively, 350, 220, and
320 vehicles per minute (veh/min). The O–D demand follows the lognormal distribution
with a mean of 1000 flow units and a variance-to-mean ratio (VMR) of 10. Note that
although the VMR seems large, the corresponding coefficient of variation (COV) is only
0.10. The SPE of a unit travel time is assumed to follow N(0, 0.8).
Effect on individual perceived travel time distribution
Effect on the perceived travel time distribution
First, we use the distribution fitting method discussed in ‘‘Fitting Distribution by
Moments’’ section to graphically depict the individual actual and perceived TTDs. Without
loss of generality, we consider a non-equilibrium route flow pattern of (400, 200, and 400
veh/min). From Fig. 3, one can see the actual and perceived TTDs could be significantly
different for all three routes. This discrepancy is attributed to the consideration of SPE,
which is conditioned on the actual TTD. Here, the SPE variance is 0.8, which makes the
perceived TTD much more dispersed and random compared to the actual TTD.
In order to further examine the effect of SPE variance on the perceived TTD, we set it at
0.0, 0.2, and 0.8, respectively. Their corresponding CDFs are shown in Fig. 4. In this study,
we only consider risk-averse and risk-neutral travelers, i.e., their confidence levels are all
greater than or equal to 0.5. Recent empirical studies (e.g., Small et al. 1999; Lam 2000;
Brownstone et al. 2003; Liu et al. 2004; de Palma and Picard 2005; Cambridge Systematics
2003; FHWA 2004,2006) revealed that most travelers are actually risk-averse. They are
keen on punctual arrivals and willing to pay a premium to avoid congestion and also
10 15 20 25 30 35 40 45
0
0.1
0.2 Route 1 actual travel time
perceived travel time
10 15 20 25 30 35 40 45
0
0.2
0.4
Probability Density
Route 2
10 15 20 25 30 35 40 45
0
0.1
0.2
Route Travel Time (min)
Route 3
Fig. 3 Probability distributions of the actual and perceived travel times
532 Transportation (2013) 40:525–548
123
to minimize the associated risk. This consideration is also widely used by many other
researches, e.g., Chen and Zhou (2009,2010), Chen et al. (2011), Lam et al. (2008), Lo
et al. (2006), Shao et al. (2006a,b,2008), Siu and Lo (2006,2008), and Zhou and Chen
(2008). Thus, the following analyses only focus on the part of distribution above the
cumulative probability of 0.5.
15 20 25 30 35 40 45
0
0.2
0.4
0.6
0.8
1
Route Travel Time (min)
Cumulative Probability
Route 1
var=0.0 (actual)
var=0.2 (perceived)
var=0.8 (perceived)
90%
15 20 25 30 35 40 45
0
0.2
0.4
0.6
0.8
1
Route Travel Time (min)
Cumulative Probability
Route 2
var=0.0 (actual)
var=0.2 (perceived)
var=0.8 (perceived)
90%
10 15 20 25 30 35 40 45
0
0.2
0.4
0.6
0.8
1
Route Travel Time (min)
Cumulative Probability
Route 3
var=0.0 (actual)
var=0.2 (perceived)
var=0.8 (perceived)
90%
Fig. 4 Actual and perceived
TTDs under different SPE
variances
Transportation (2013) 40:525–548 533
123
From Fig. 4, we can observe that:
a. With the increase of SPE variance, the perceived TTDs gradually move to the right
with a larger variability. This means that for a given cumulative probability (e.g.,
90 %), the perceived travel time (PTT) is increasing with the SPE variance.
b. For each route, the increase value of PTT is diverse. For example, routes 2 and 3 have
the largest and smallest increase of PTT. This change may result in an adjustment of
travelers’ PTT distribution-based route choice decisions.
Effect on the components of perceived mean-excess travel time
Second, we examine the effect of SPE on the components of PMETT. As shown in Fig. 2,
the PMETT can be considered as a sum of three components: PMT, PBT, and PEED. The
sum of the first two components equals the PTTB. The PTTB and PEED represent the
reliability and unreliability aspects of the perceived travel time variability, respectively.
For demonstration purpose, we only show route 1 in Fig. 5. Here, we assume all travelers
have the same confidence level of 85 %.
From Fig. 5, one can see that:
a. The SPE variance has no effect on the PMT while its mean has a significant effect on
the PMT.
b. With the increase of mean and variance, the sum of PBT and PEED is increasing. This
means if travelers’ perception on network conditions becomes more inaccurate, a larger
amount of time is needed (i.e., added to the mean travel time) in order to ensure on-time
arrival at a given confidence level and also to avoid excessively late trips.
c. For the PBT, the effect of variance is larger than that of the mean. For the PEED, the trend is
opposite. These results seem to imply that the SPE variance (mean) plays a more important
role in the reliability (unreliability) aspect of the perceived travel time variability.
d. The magnitudes of PBT and PEED are similar, suggesting the unreliability aspect has
a commensurate importance as the reliability aspect in travelers’ route choice
decisions. Thus, it is indeed necessary to simultaneously consider both reliability and
unreliability aspects of the perceived travel time variability in making route choice
decisions under uncertainty.
Effect on route choice decision
In this experiment, we examine the effect of SPE at the route choice decision level.
Specifically, the equilibrium flows and PMETTs under different combinations of SPE and
confidence level (CL) aare analyzed.
37.0873 3.6272 8.3835
31.3815 4.0150 6.9787
31.3815 3.1929 7.0837
25 30 35 40 45 50
C
B
A
N
(0.1, 0.1)
N
(0.1, 0.3)
N
(0.3, 0.1)
PMT PBT PEED
Fig. 5 Effect of perception error on the components of PMETT on route 1
534 Transportation (2013) 40:525–548
123
First, we compare the equilibrium results of METE (without perception error) and
SMETE (with SPE) models in Table 2. As expected, both the equilibrium conditions and
conservation constraints are fully satisfied. In addition, the difference between the equi-
librium METT and PMETT is significant, indicating the importance of considering trav-
elers’ perception error. Note that route 2 has the largest difference between the METT and
PMETT for both models among the three routes. The main reason is that the SPE is
conditioned on the actual travel time distribution. From Table 2, one can see route 2 has
the largest mean travel time and a medium standard deviation (SD). A lower confidence
level (CL) of 0.7 results in a more dominant role of mean travel time in the METT/
PMETT. Hence, the effect of SPE is more significant on route 2.
Second, we investigate the effects of SPE and CL on route choice decisions (in terms of
the equilibrium PMETTs and flows). The CLs of 0.5, 0.7, and 0.9 are used to represent
travelers’ different risk attitudes. In each panel of Figs. 6and 7, the leftmost point with
Table 2 Equilibrium results of the METE and SMETE models (CL =0.7)
Model Route # Flow (veh/min) Cost (min)
Mean SD METT PMETT Difference
METE 1 371.53 26.91 3.58 30.62 38.99 8.37
2 220.02 28.70 4.79 30.62 40.31 9.69
3 408.45 24.83 5.38 30.62 38.25 7.63
SMETE (N(0.2, 0.6)) 1 373.46 32.42 5.94 30.79 39.17 8.38
2 211.77 33.69 6.56 29.33 39.17 9.84
3 414.77 30.36 7.83 31.43 39.17 7.74
0.2 0.4 0.6 0.8 1.0
25
30
35
40
45
50
55
60
alpha=0.5
PMETT(min)
0.2 0.4 0.6 0.8 1.0
25
30
35
40
45
50
55
60
alpha =0.7
Var. of Perception Error
0.2 0.4 0.6 0.8 1.0
25
30
35
40
45
50
55
60
alpha=0.9
mea n=0. 0
mea n=0. 1
mea n=0. 2
mea n=0. 3
mea n=0. 4
PMETT=39.17
METT=3 0 . 6 2
Fig. 6 Equilibrium PMETTs under different SPEs and CLs
Transportation (2013) 40:525–548 535
123
zero-mean and zero-variance corresponds to the METE model without perception error.
From these results, the following observations can be drawn:
•With the increase of SPE mean and variance, the equilibrium PMETT is always
increasing. Travelers add more time to ensure the reliability and unreliability
requirements under travel time variability and perception error. This result indicates
that ignoring travelers’ perception error will affect their long-term trip time planning
(e.g., departure time choice, arrival time estimation).
•The SPE has a more significant effect on the route choice decisions for the more risk-
averse travelers with a higher confidence level. Perhaps this is because the SPE adds
extra uncertainty to the actual travel time variability, and the more risk-averse travelers
will accordingly add a larger PBT and PEED to hedge against the perceived travel time
variability. This will consequently result in a larger adjustment of route choice
decisions as indicated by the different changing trend of the equilibrium route flow
pattern in the right panels of Fig. 7.
Effect on total travel time distribution
In this section, we investigate the effect of SPE on the total travel time (TTT) distribution.
We show the TTT distributions corresponding to the system optimal (SO), METE, and
SMETE models in Fig. 8. For demonstration purpose, we assume the SPE follows N(0.2,
0.6) and travelers’ CL is 0.7. Note that Chen et al. (2007) found mean and variance are not
appropriate measures for describing the asymmetric TTT distribution and they recom-
mended using the concept of total travel time budget (TTTB). TTTB is essentially the
percentile of TTT distribution at a given reliability requirement. Thus, we also show the
TTTBs at 0.5, 0.7 and 0.9 percentiles corresponding to the SO (a lower bound for
0.2 0.4 0.6 0. 8 1.0
360
365
370
375 Route 1 (alp ha=0.5)
Var. of Percep tion Error
Flow (veh/min)
0.2 0.4 0.6 0. 8 1.0
370
372
374
376 Route 1 (alp ha=0.7)
Va r. of P er c ep tio n E rr o r
Flow (veh/min)
0.2 0.4 0.6 0. 8 1.0
388
390
392
394 Route 1 (alpha= 0.9)
Var. of Percep tion Error
Flow (veh/min)
0.2 0.4 0.6 0. 8 1.0
200
210
220
230 Route 2 (alpha= 0.5)
Var. of Percep tion Error
Flow (v eh/min)
0.2 0.4 0.6 0. 8 1.0
200
210
220
230 Route 2 (alpha= 0.7)
Var. of Percep tion Error
Flow (veh/min)
0.2 0.4 0.6 0. 8 1.0
194
196
198
200 Route 2 (alp ha=0.9)
Var. of Percep tion Error
Flow (v eh/min)
0.2 0.4 0.6 0. 8 1.0
410
415
420
425 Route 3 (alpha= 0.5)
Va r. of P er c e
p
tion Error
Flow (veh/min)
0.2 0.4 0.6 0. 8 1.0
405
410
415
420 Route 3 (alpha= 0.7)
Va r. of P er c e
p
tion Error
Flow (veh/min)
0.2 0.4 0.6 0. 8 1.0
411.5
412
412.5 Ro ute 3 (alpha=0. 9)
Va r. of P er ce
p
tion Error
Flow (veh/min)
mea n=0.0 me an=0. 2 mean= 0.4
Fig. 7 Equilibrium flows under different SPE distributions and CLs
536 Transportation (2013) 40:525–548
123
estimating the TTT distribution), METE (without perception error), and SMETE models in
this figure. Here, we simply define the TTTB difference between METE/SMETE and SO
models as an efficiency loss measure. From Fig. 8, we can observe that:
a. The TTT distribution corresponding to the METE/SMETE-based flow pattern is
different from that corresponding to the SO-based flow pattern. This means an
efficiency loss exists when using the METE/SMETE model. Meanwhile, the efficiency
loss is increasing with the reliability requirement.
b. We pay attention to the distribution tails, where SMETE is above METE and METE is
above SO. This relationship shows that the SMETE-based TTT distribution has the
largest variance and asymmetry. The large variance is due to the consideration of SPE,
which adds extra uncertainty to the total travel time variability. The large asymmetry
indicates the necessity of using the TTTB as a risk measure for assessing network
performance more systematically.
c. When considering travelers’ SPE on network conditions (i.e., SMETE), the efficiency
loss also becomes larger. Thus, ignoring the SPE in the risk-averse traffic equilibrium
models can lead to bias estimation of the TTT distribution, resulting in an inaccurate
network performance assessment.
Finally, we use the well-known Nguyen-Dupuis network to further verify the above
results. The Nguyen-Dupuis network (1984), shown in Fig. 9, consists of 13 nodes, 19
links, and 4 O–D pairs. We use the standard BPR function with the deterministic link free-
flow travel time and capacity shown in Yin et al. (2009). The random demand is assumed
to follow the lognormal distribution. The expected demands of O–D pairs (1, 2), (1, 3), (4,
2), and (4, 3) are respectively 200, 400, 300, and 100, and the VMRs are all equal to 2.0,
corresponding to the COVs of 0.10, 0.07, 0.08, and 0.14. For demonstration purpose, all
travelers are assumed to have the same confidence level of 0.80 and the SPE follows N(0.2,
0.4). From Fig. 10, one can observe similar relationship of TTT distributions among
the SO, METE, and SMETE models as shown in Fig. 8. Again, the TTT distribution
corresponding to the METE-based flow pattern is obviously different from that of the
22.5 3 3.5
x 10
4
0
0.5
1
1.5
2
2.5
x 10
-4
Total Travel Time
Probability Density
SO
METE
SMETE
Reliability Probability
TTTB 0.5 0.7 0.9
SO 25506 26285 27492
METE 25894 26740 28067
SMETE 26022 26886 28246
METE-SO 388 455 575
SMETE-SO 516 601 754
Fig. 8 TTT distributions under different equilibrium models
Transportation (2013) 40:525–548 537
123
SMETE-based flow pattern. When considering travelers’ perception error on network
conditions, the TTT becomes larger and more random.
Discussions
Interpreting stochastic perception error distribution
We interpret the SPE mean and variance in terms of traveler’s characteristics (e.g.,
familiarity of network conditions) and trip purposes. For simplicity, travelers are classified
as four types according to the mean and variance: (1) taxi drivers with very accurate
perception on network conditions (i.e., have knowledge of both available routes and their
Fig. 9 Nguyen-Dupuis network
33.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4
x 10 4
0
1
2
3
4
5x 10 -4
Total Travel Time
Probability Density
SO
METE
SMETE
Fig. 10 TTT distributions under different models (Nguyen-Dupuis network)
538 Transportation (2013) 40:525–548
123
respective TTDs), (2) regular commuters with keen perception, (3) regular commuters with
median perception, and (4) unfamiliar drivers with little knowledge about network con-
ditions. For demonstration purpose, we only show route 1 in Fig. 11.
Taxi drivers
Taxi drivers are usually very familiar with network conditions due to their rich experi-
ences, and thus can perceive network conditions very accurately. In this scenario, we use a
zero-mean and a small SPE variance (e.g., 0–0.03) as shown in Fig. 11a to model taxi
drivers’ perception on network conditions. We can observe that the perceived TTD is
almost identical to the actual TTD. Accordingly, the PMETT is also close to the actual
METT.
Regular commuters with keen perception
Regular commuters, especially those who commute between a fixed O–D (e.g., between
home and work place) for many years, are also very familiar with the network conditions
of their commuting routes. We can model their perception error as follows: the variance is
similar to that of taxi drivers but they may add a ‘safety margin’ time to the actual travel
time. For different trip purposes, the ‘safety margin’ time could be different. We discuss
three typical trip purposes as follows:
•Work trips. For these trips, regular commuters with keen perception could perceive
network conditions accurately due to their day-to-day commuting experiences.
20 25 30 35 40 45
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
Route Travel Time (min)
Probability Density
var=0.00 (actual)
var=0.01 (perceived)
var=0.03 (perceived)
Actual travel time
(a) Taxi drivers
20 25 30 35 40 45
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
Route Travel Time (min)
Probability Density
mean=0.00, var=0.00 (actual)
mean=0.02, var=0.00 (perceived)
mean=0.04, var=0.00 (perceived)
mean=0.06, var=0.00 (perceived)
mean=0.08, var=0.00 (perceived)
Regular commuters with keen perception
(b)
20 25 30 35 40 45 50
0
0.05
0.1
0.15
0.2
Route Travel Time (min)
Proba bility Density
mean=0.00, var=0.00 (actual)
mean=0.10, var=0.02 (perceived)
mean=0.20, var=0.02 (perceived)
Regular commuters with median perception
20 25 30 35 40 45 50 55 60 65 70
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
Route Travel Time (min)
Probability Density
mean=0.00, var=0.00 (actual)
mean=0.50, var=0.20 (perceived)
mean=0.50, var=0.40 (perceived)
mean=0.50, var=0.60 (perceived)
mean=0.50, var=0.80 (perceived)
(d) unfamiliar drivers
(c)
Fig. 11 Perceived travel times distributions of different types of travelers
Transportation (2013) 40:525–548 539
123
The case with a mean of 0.02 in Fig. 11b may mimic this type of trips. The horizontal
offset between the perceived and actual TTDs is very small.
•Trips for important events. Even though some commuters are familiar with network
conditions, they still allocate a relative large ‘safety margin’ time for some important
events (e.g., going to airport, job interview, important meeting). The reason is that the
penalty is extremely high when they cannot punctually arrive at the destination. The
mean of 0.08 in Fig. 11b may correspond to these trips, where the gap between the
actual and perceived TTDs is larger than that of the work trips.
•Leisure trips. For the leisure trips, the on-time arrival requirement is not as important;
therefore, we can use the mean of 0.04–0.06 to mimic them.
Regular commuters with median perception
Note that not all regular commuters are familiar with network conditions. Their knowledge
of network conditions is dependent on both the experience they obtained from traveling
and their curiosity of seeking travel information. Generally, if a commuter does not go to a
fixed working destination for a long time, they may not have accurate perception of
network conditions. Also, if a commuter does not have a strong curiosity of seeking travel
information, he/she may not know the travel time ranges of the routes available for his/her
trips, even though he/she travels to a fixed working destination for a certain period of time.
We can use a small/median mean and variance to model the perception of this type of
commuters. Figure 11c shows there are obvious differences between the actual and
perceived TTDs. In addition, different mean values only result in a horizontal shift of the
perceived TTD while the other characteristics remain almost the same.
Unfamiliar drivers
For the drivers who are not familiar with network conditions (e.g., tourists), they may have
a limited knowledge on the routes available for their trips. We may use a large mean and a
large variance as shown in Fig. 11d to mimic their lack of knowledge of network condi-
tions. One can see the actual and perceived TTDs are significantly different. In addition,
with the increase of SPE variance, the perceived TTD gradually becomes more flat. For
this type of travelers, in order to improve their traveling experience, an advanced traveler
information system (ATIS) or a route guidance system (RGS) is indispensable.
Recall that Fig. 10 shows the relationship of TTT distributions among the SO, METE,
and SMETE models. Now we continue to use the Nguyen-Dupuis network to further
examine the effect of four traveler types according to the mean and variance of the SPE
distribution on network-wide performance. Specifically, we demonstrate the TTT distri-
butions in Fig. 12 when network users are regular commuters with keen perception
(mean =0.04, var =0), regular commuters with median perception (mean =0.20,
var =0.02), and unfamiliar drivers (mean =0.50, var =0.80), respectively. The TTT
distribution for the taxi drivers is essentially similar to the METE model (i.e., no per-
ception error or very accurate perception as shown in Fig. 11a). With these TTT distri-
butions, we can see significant differences among network users with different SPE
distributions. When the network is dominated by travelers with less knowledge of network
conditions, the TTT becomes larger and more random as indicated by the unfamiliar
drivers in Fig. 12. These TTT distributions, albeit homogeneous in each user class in terms
540 Transportation (2013) 40:525–548
123
of the parameters used to simulate stochastic perception error, serve as a benchmark that
can be used to assess any heterogeneous user-class combinations.
Network impacts
From the network modeling (planners’) perspective, it is necessary to classify network
travelers in terms of not only their characteristics on the knowledge (or familiarity) of
network conditions (e.g., taxi drivers, regular commuters, and unfamiliar drivers), but also
their trip purposes (e.g., work trips, trips for important events, and leisure trips). In
addition, an explicit consideration of multiple user classes in terms of confidence levels
(CLs) is also important. Different CLs represent different risk attitudes towards the per-
ceived travel time variability. These treatments may enhance the modeling realism of path
finding models and traffic equilibrium models under uncertainty (e.g., Chen and Ji 2005;
Lo et al. 2006).
From the network travelers’ perspective, travel time information provision can alleviate
the negative impact of travel time variability such that the ‘scheduled/budgeted’ travel time
and the risk of encountering excessively late arrivals can both be reduced. Equipping an
ATIS or a RGS for unfamiliar drivers (e.g., tourists) is particularly indispensable in order
to improve their traveling experience in a new environment. For commuters, frequent
travel time information provided by the transportation information center via variable
message signs, Internet, or other medias is helpful in making informed travel choice
decisions.
From the comparison of the actual and perceived travel time distributions, we have
similar findings as previous studies. Particularly, Ben-Elia et al. (2008) conducted simu-
lated experiments to examine the combined effect of information provision and personal
experience on route choice decisions under travel time variability. They also found that
unfamiliar drivers will benefit the most from the information while the benefit is limited for
3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9
x 104
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5 x 10-4
Total Travel Time
Probability Density
METE
SMETE (mean=0.04, var=0.00)
SMETE (mean=0.20, var=0.02)
SMETE (mean=0.50, var=0.80)
Regular commuters
with keen perception
Unfamiliar
drivers
Regular commuters
with median perception
Fig. 12 TTT distributions of different traveler types
Transportation (2013) 40:525–548 541
123
familiar drivers (in particular, taxi drivers and commuters with keen perception). In
addition, the information provision plays different roles for trips with different purposes.
Moreover, Ben-Elia et al. (2008) examined the effect of travel time information provision
(or enhancing the travelers’ perception) on risk attitudes towards uncertainty. They con-
cluded that information reduces initial exploration, increases initial risk seeking, and
contributes to between-subjects’ differences in responding to travel time variability. These
findings also support our suggestions to network modelers, i.e., the necessity of classifying
and surveying travelers in terms of their characteristics on the knowledge of network
conditions and trip purposes, as well as their risk attitudes towards travel time variability
with different confidence levels. However, as noted by Ettema and Timmermans (2006), it
is difficult to quantitatively assess the benefit of informing travelers about travel time
variability.
Even with the route guidance information, the gap between travelers’ perceived and
actual travel time variability, however, depends on the information quality (e.g., how
frequent the information is updated) and whether the travelers comply with the provided
information. Thus, it is also necessary to explicitly consider the information quality and
travelers’ adoption/compliance with the information in the traffic equilibrium models
under uncertainty (e.g., Yang and Huang 2004; Yang and Meng 2001).
Conclusions and future research
In this study, we investigate the effects of stochastic perception error (SPE) in the stochastic
mean-excess traffic equilibrium model at three levels: (1) individual perceived travel time
distribution as well as its connection to the classification by types of travelers and trip
purposes, (2) perceived mean-excess travel time (PMETT)-based route choice decisions (in
terms of equilibrium flows and PMETTs), and (3) network performance measure (in terms of
the total travel time distribution and its statistics). In all three levels, a curve fitting method is
adopted to estimate the whole distribution of interest based on its first four moments.
Numerical examples are also presented to illustrate and visualize the above analyses.
The paper is a network model-driven analysis of the effects of SPE under travel time
variability. The actual and perceived travel time distributions (or moments) are explicitly
derived from the travel demand uncertainty and SPE distributions. This analysis is thus
different from the empirical data-based statistical studies on the perception of travel time
unreliability using stated preference surveys.
The analysis results indicate the importance of explicitly considering the SPE in the
traffic equilibrium models under uncertainty. For travelers’ individual route choice deci-
sions, ignoring the SPE will affect their travel route selection and trip time planning
(e.g., departure time choice, arrival time estimation). This effect is especially obvious for
the drivers with limited knowledge on network conditions and for trips with important
purposes. For network planners and modelers, ignoring the SPE in the traffic equilibrium
models under uncertainty can lead to bias estimations of equilibrium traffic flows, which
consequently results in an inaccurate network performance assessment. In addition, it is
necessary to classify travelers and then collect data according to their characteristics on
network conditions and trip purposes as well as their risk attitudes towards travel time
variability with different confidence levels. These classifications will provide more detailed
perception and travel time variability data, further enhancing the modeling realism of path
finding models and traffic equilibrium models under uncertainty (e.g., Chen and Ji 2005;
Lo et al. 2006).
542 Transportation (2013) 40:525–548
123
For future research, several directions are worthy of further investigation:
(1) In this study, we use the normal distribution to characterize the SPE. Calibration of
the distribution parameters is thus a critical issue. Empirical data on travelers’
perception will be used to verify this assumption and also to compare the fitting
quality with other distributions. To carry out this task, a practical and economical
perception data collection method is especially important.
(2) Note that Fujii and Kitamura (2004) verified the hypothesis that ‘‘drivers perceive
uncertain travel time as an interval’’. To cater for different data preparations, we can
also use other uncertainty modeling philosophies such as the fuzzy variable or hybrid
(random-fuzzy) variable to characterize the perception error of travel time variability.
(3) The correlations of traffic flows and/or travel times should be considered. The
concept of copula can be a potential valuable tool to deal with this issue.
(4) Other appropriate travel time distributions (e.g., Gamma, Weibull, and Burr) and
more robust curve fitting methods will also be explored to enhance the distribution
fitting quality.
(5) We will test on different-sized networks with different traffic conditions to compare
the findings reported in this paper.
(6) Theoretically the current modeling framework (i.e., the SMETE model) can capture
all risk attitudes. However, it may not work for risk-prone travelers in implemen-
tation. The reason is that the METT requires the TTB to be positive, which is not
always guaranteed for risk-prone travelers. To enhance the modeling flexibility, we
need to develop a totally new model in the future. This new model should have a
complete characterization of risk attitudes toward travel time variability (i.e., risk-
prone, risk-neutral, and risk-averse travelers) and also a complete characterization of
travel time variability (i.e., both reliability and unreliability requirements of travelers)
simultaneously.
Acknowledgments The authors are grateful to Prof. Patricia Mokhtarian (North America Co-editor of
Transportation) and two referees for providing constructive comments and suggestions for improving the
quality and clarity of the paper. The work of the first author was supported by the China Scholarship
Council, and the work of the second author was supported by a CAREER grant from the National Science
Foundation of the United States (CMS-0134161), and an Oriental Scholar Professorship Program sponsored
by the Shanghai Ministry of Education in China.
Appendix 1
This appendix provides the equivalent variational inequality (VI) formulation of the
SMETE conditions. First, a feasible route flow pattern fshould satisfy the following basic
conditions:
X
p2Pw
fw
p¼qw;8w2W;ð12Þ
va¼X
w2WX
p2Pw
fw
pdw
pa;8a2A;ð13Þ
fw
p0;8p2Pw;w2W;ð14Þ
where v
a
is the flow on link a;dw
pa=1 if route pconnecting O–D pair wuses link a, and 0,
otherwise. Eq. (12) is the travel demand conservation constraint; Eq. (13) is a definitional
constraint that sums up all route flows that pass through a given link; and Eq. (14)is
Transportation (2013) 40:525–548 543
123
a non-negativity constraint on the route flows. In the following, we use Xto denote the
constraint set that consists of Eqs. (12)–(14).
The SMETE conditions can be equivalently formulated as the following variational
inequality (VI) problem, which is to find a route flow pattern f2X, such that
~
gf
ðÞ
Tff
ðÞ0;8f2Xð15Þ
For the existence of the equilibrium route flow pattern and the equivalence between the
SMETE conditions and the solution to the above VI problem, interested readers may refer
to Chen and Zhou (2009) and Chen et al. (2011).
Appendix 2
This appendix provides a detailed derivation of the probability distribution statistics for the
lognormal distributed travel demand uncertainty. Chen and Zhou (2009) assumed the
uncertainty only comes from the random free-flow travel time. As is well-known, day-to-
day demand fluctuation and link capacity degradation are two main uncertainty sources in
the transportation system. Also, most of the recent literature considered either demand
uncertainty (e.g., Chen and Zhou 2010; Chen et al. 2011; Shao et al. 2006a,b) or capacity
uncertainty (e.g., Lo et al. 2006; Siu and Lo 2006), or both (e.g., Lam et al. 2008; Shao
et al. 2008; Siu and Lo 2008) in the traffic equilibrium models under uncertainty. In this
study, we consider the day-to-day demand fluctuation as a representative uncertainty
source. Following Zhou and Chen (2008), the lognormal distribution is adopted to char-
acterize the demand uncertainty. It is a nonnegative, asymmetrical distribution and has
already been adopted as a more realistic approximation of the fluctuated travel demand to
investigate the uncertainty propagation in the four-step travel demand forecasting proce-
dure (Zhao and Kockelman 2002).
The mean and variance of the lognormal distributed travel demand between a generic
O–D pair ware denoted as q
w
and e
w
, respectively. Consider the commonly used BPR
(Bureau of Public Roads) function:
Ta¼t0
a1þhVa
Ca
b
"#
;ð16Þ
where T
a
and V
a
are the random link travel time and flow, respectively; t0
aand C
a
are,
respectively, the deterministic link free-flow travel time and capacity; hand bare BPR
parameters. Then, the probability distribution or statistics of flow and travel time variables
can be derived as shown in Table 3. One can see the actual and perceived travel time
distributions are explicitly derived from the travel demand and SPE distributions. This
approach is different from the empirical studies on the perception of travel time unreli-
ability using stated preference surveys (e.g., Tseng et al. 2009).
Using the Cornish and Fisher (1937)’s asymptotic expansion, we can estimate the PTTB
at a user-specified confidence level aas
~
nw
paðÞCum ~
Cw
p
1
þww
paðÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Cum ~
Cw
p
2
s;ð17Þ
where ww
paðÞis related to the skewness and kurtosis of the perceived TTD. It can be
calculated as follows:
544 Transportation (2013) 40:525–548
123
ww
paðÞ¼U1aðÞþ1=6ðÞU1aðÞ
21
hi
S~
Cw
p
hi
þ1=24ðÞU1aðÞ
33U1aðÞ
hi
K~
Cw
p
hi
1=36ðÞ2U1aðÞ
35U1aðÞ
hi
S~
Cw
p
hi
2;
ð18Þ
Table 3 Statistics of flow and travel time variables
Random variable Statistics
Travel demand QwQwLN lw;q;rw;q
;
lw;q¼ln qw
ðÞ
1
2ln 1 þew.qw
ðÞ
2
;rw;q
2¼ln 1 þew.qw
ðÞ
2
:
Route flow Fw
pFw
pLN lw
p;f;rw
p;f
;
lw
p;f¼ln fw
p
1
2ln 1 þew=qw
fw
p
!
;rw
p;f
2¼ln 1 þew=qw
fw
p
!
:
Link flow VaVaLN la;v;ra;v
;
la;v¼ln va
1
2ln 1 þea;v.va
2
;ra;v
2¼ln 1 þea;v.va
2
;
va¼P
wP
p
fw
pdw
pa,ea;v¼P
wP
p
fw
pew
qwdw
pa,
EV
a
ðÞ
n
½
¼exp nla;v
þn2
2ra;v
2
hi
;n¼1;2;3;4.
Actual link travel time TaET
a
ðÞ
n
½¼t0
a
nP
n
i¼0
n!
i!niðÞ!hi
Ca
ðÞ
biEV
a
ðÞ
bi
hi
;n¼1;2;3;4,
CM Ta
ðÞ
n
½¼ET
aET
a
ðÞðÞ
n
½;n¼2;3;4,
Cum Ta
ðÞ
i
¼ET
a
ðÞ
i
;i¼1;2;3,
Cum Ta
ðÞ
4
hi
¼CM Ta
ðÞ
4
hi
3CM Ta
ðÞ
2
hi
2.
Perceived link travel time ~
TaE~
Ta
¼1þlðÞET
a
½,E~
Ta
2
hi
¼1þlðÞ
2ET
a
ðÞ
2
hi
þr2ET
a
½,
E~
Ta
3
hi
¼1þlðÞ
3ET
a
ðÞ
3
hi
þ31þlðÞr2ET
a
ðÞ
2
hi
,
E~
Ta
4
hi
¼1þlðÞ
4ET
a
ðÞ
4
hi
þ61þlðÞ
2r2ET
a
ðÞ
3
hi
þ3r4ET
a
ðÞ
2
hi
,
CM ~
Ta
n
¼E~
TaE~
Ta
n
;n¼2;3;4,
Cum ~
Ta
i
hi
¼E~
Ta
i
hi
;i¼1;2;3,
Cum ~
Ta
4
hi
¼CM ~
Ta
4
hi
3CM ~
Ta
2
hi
2.
Actual route travel time Cw
pCum Cw
p
n
hi
¼Pa2ACum Ta
ðÞ
n
½dw
ap;n¼1;2;3;4.
Perceived route travel time ~
Cw
pCum ~
Cw
p
n
hi
¼Xa2ACum ~
Ta
n
dw
ap;n¼1;2;3;4;
S~
Cw
p
hi
¼Cum ~
Cw
p
3
,Cum ~
Cw
p
2
1:5
;
K~
Cw
p
hi
¼Cum ~
Cw
p
4
,Cum ~
Cw
p
2
2
:
CM[.] central moments, Cum[.] cumulants, S[.] skewness, K[.] kurtosis. LN lognormal distribution, Nl;r2
ðÞ
is the SPE distribution of a unit travel time
Transportation (2013) 40:525–548 545
123
where U1ðÞis the inverse of the standard normal cumulative distribution function (CDF);
S~
Cw
p
hi
and K~
Cw
p
hi
are the skewness and kurtosis of ~
Cw
pfor quantifying the asymmetry and
peakedness of the probability distribution of ~
Cw
p, respectively.
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Author Biographies
Xiangdong Xu is completing his Ph.D. studies in Transportation Planning and Management in the
Department of Transportation Engineering at Southeast University, China. His research interest mainly lies
in network equilibrium modeling and algorithmic development, modeling risk-taking route choice behavior,
and transportation network design under uncertainty.
Anthony Chen is a Professor in the Department of Civil and Environmental Engineering at Utah State
University. The major portion of his research focuses on transportation systems modeling and analysis,
transportation network reliability/vulnerability/resiliency analysis, origin–destination trip table estimation,
network modeling and solution algorithmic development, and non-motorized transportation modeling.
Lin Cheng is a Professor in the Department of Transportation Engineering at Southeast University, China.
His research mainly includes transportation network reliability analysis, solution algorithmic development
of traffic equilibrium models, transportation network bottlenecks identification, transportation network
simulation, and geographic information system.
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